Minimal percolating sets for mutating infectious diseases

This paper is dedicated to the study of the interaction between dynamical systems and percolation models, with views toward the study of viral infections whose virus mutate with time. Recall that [Formula: see text]-bootstrap percolation describes a deterministic process where vertices of a graph are infected once [Formula: see text] neighbors of it are infected. We generalize this by introducing [Formula: see text]-bootstrap percolation, a time-dependent process where the number of neighboring vertices that need to be infected for a disease to be transmitted is determined by a percolation function [Formula: see text] at each time [Formula: see text]. After studying some of the basic properties of the model, we consider smallest percolating sets and construct a polynomial-timed algorithm to find one smallest minimal percolating set on finite trees for certain [Formula: see text]-bootstrap percolation models.